Abstract. Different ice thickness distributions along the flow line and the flow line length changes of the Gregoriev Ice Cap, Terskey Ala-Tau, Central Asia, were obtained for some surface mass balance histories which can be considered as possible surface mass balances in the future. The ice cap modeling was performed by solving of steady state hydrodynamic equations in the case of low Reynolds number in the form of the mechanical equilibrium equation in terms of stress deviator components coupled with the continuity equation for incompressible fluid. The numerical solution was obtained by the finite difference method. A compound approximation of the ice surface boundary condition based on the extending of the mechanical equilibrium equation to ice surface points was applied. The approximation is considered as a way to overcome the problem of diagnostic equations numerical solution stability in the full model.

The basal sliding can arise in the glacier tongue at certain climatic conditions and was introduced both through linear and through non-linear friction laws.

A possible glacier length history, that corresponds to the regional climate changes derived from the tree-rings data, was obtained by the model.

The correlations between the glacier length changes and annual air temperature histories were investigated within the simplified equation introduced by J. Oerlemans in the form of linear dependence of annual air temperature versus glacier length and time derivative of the length. The parameters of the dependence were derived from modeled glacier length histories that correspond to harmonic climate histories. The parameters variations were investigated for different periodicities of harmonic climate histories and appropriate dependences are presented in the paper. The results of the modeling are in a good agreement with the J. Oerlemans climatic model.